Finite Group Databasebeta

← #10692430

Subgroups — #10692430

Order 768.

420 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83319C211yesmaximal, characteristic
2#83280C211yesmaximal, characteristic
3C3×#1551C211yesmaximal, characteristic
4#70815C211yesmaximal, characteristic, complemented
5#70429C211yesmaximal, characteristic, complemented
6#67757C211yesmaximal, characteristic, complemented
7#67708C211yesmaximal, characteristic, complemented
8#9373—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13#5239—22no—
14#5234—22no—
15#5090—21no—
16#5075—21no—
17#5060C412yes—
18#5055C412yes—
19#5012—21no—
20#5011—22no—
21D4×C3⋊C8—21no—
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24#4925C2211yescharacteristic
25C3×#333—21no—
26C3×#331—21no—
27#4484—21no—
28#4484—21no—
29#4465—22no—
30#3216—31no—
31#2981—31no—
32#1551D311yescharacteristic
33#1537—31no—
34#1481—31no—
35#1290—31no—
36#1287—31no—
37C2×C3×#41D412yes—
38C22×C3×C8—21no—
39C3×C4⋊Q8D411yescharacteristic
40C3×#127D411yescharacteristic
41C3×#124C2×C412yes—
42C3×#121—21no—
43C3×#121—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C2×C3×C42C2311yescharacteristic
47C2×#216—22no—
48C2×#214—22no—
49#926—21no—
50#924—21no—
51#924—21no—
52C2×#208—22no—
53C22×C3⋊C8—22no—
54#870—21no—
55#869C2×C411yescharacteristic
56#869C2×C411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×#97—21no—
64#835—22no—
65#832—22no—
66#830—22no—
67#830—22no—
68#830—21no—
69#830—21no—
70#830—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74#803C2×C412yes—
75#803D412yes—
76C4×C3⋊C8D411yescharacteristic
77C4×C3⋊C8D411yescharacteristic
78#533—31no—
79#532C2×D311yescharacteristic
80#497—32no—
81#489—32no—
82#462—61no—
83#460—61no—
84#452—62no—
85#436—61no—
86#435—62no—
87C8×D4—61no—
88#432—31no—
89#432—31no—
90#432C2×D311yescharacteristic
91#431—31no—
92C2×C4⋊C8C2×D311yescharacteristic
93#408—62no—
94#333—21no—
95#331—21no—
96#330—62no—
97Q8⋊C8—61no—
98Q8⋊C8—61no—
99C3×#41—82no—
100C2×C3×Q8C22⋊C412yes—
101C2×C3×D4—22no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4C22⋊C412yes—
105C22×C3×C4—22no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C2×Dic6—21no—
110C2×Dic6—21no—
111C2×C3×C8—41no—
112C2×C3×C8—41no—
113C2×C3×C8—81no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C22⋊C4—22no—
117C3×C22⋊C4—22no—
118C3×C42C22×C411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122#217—41no—
123#216—82no—
124#214—82no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129Dic3⋊C4—41no—
130Dic3⋊C4—41no—
131C4×Dic3—41no—
132#208—82no—
133C2×C3⋊C8—22no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—21no—
136C2×C3⋊C8—22no—
137C2×C3⋊C8—82no—
138C2×#41D1212yes—
139C2×Q16—62no—
140C2×SD16—62no—
141C2×M16—62no—
142C22×C8—21no—
143C22×C8—62no—
144C4⋊Q8—31no—
145C4⋊Q8—31no—
146C4⋊Q8#6711yescharacteristic
147#127#6711yescharacteristic
148#125—61no—
149#124C4×D312yes—
150C22⋊Q8—61no—
151C22⋊Q8—61no—
152#121—21no—
153#121—21no—
154C4×Q8—61no—
155C4×Q8—61no—
156C4×D4—21no—
157C4×D4—21no—
158#117—61no—
159C2×C42C22×D311yescharacteristic
160C4⋊C8—21no—
161C4⋊C8—21no—
162C4⋊C8—21no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166#103—62no—
167#102—62no—
168C22⋊C8—21no—
169C22⋊C8—62no—
170C22⋊C8—62no—
171C22⋊C8—61no—
172C22⋊C8—61no—
173C22⋊C8—61no—
174#97—21no—
175#97—32no—
176#97—32no—
177C4×C8—31no—
178C4×C8—31no—
179C23×C3—22no—
180C23×C3#12311yescharacteristic
181C3×Q8—42no—
182C3×D4—42no—
183C3×D4—41no—
184C3×D4—41no—
185C3×D4—81no—
186C3×D4—82no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—22no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4C2×C22⋊C411yescharacteristic
195C2×C3×C4C4×D411yescharacteristic
196C2×C3×C4C4×D411yescharacteristic
197C2×C3×C4#12011yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×Dic3—41no—
201C2×Dic3—41no—
202Dic6—41no—
203Dic6—41no—
204Dic6—42no—
205C3×C8—81no—
206C3×C8—81no—
207C3⋊C8—41no—
208C3⋊C8—42no—
209C3⋊C8—41no—
210C3⋊C8—42no—
211#41—82no—
212C2×Q8—61no—
213C2×Q8—61no—
214C2×Q8(C2×D3)⋊C412yes—
215C2×D4—22no—
216C2×D4—21no—
217C2×D4—21no—
218C2×D4(C2×D3)⋊C412yes—
219C22×C4—22no—
220C22×C4C2×D1211yescharacteristic
221C22×C4#23711yescharacteristic
222C22×C4#23711yescharacteristic
223Q16—242no—
224SD16—242no—
225M16—242no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—61no—
229C2×C8—61no—
230C2×C8—62no—
231C2×C8—62no—
232C2×C8—81no—
233C2×C8—242no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C22⋊C4—121no—
245C42—121no—
246C42C2×C4×D311yescharacteristic
247C42D3×D411yescharacteristic
248C42D3×D411yescharacteristic
249C42C2×#6711yescharacteristic
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—42no—
254C22×C3—42no—
255C22×C3—42no—
256C22×C3#38611yescharacteristic
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—42no—
263C3×C4#47111yescharacteristic
264C3×C4#47311yescharacteristic
265Dic3—81no—
266Dic3—81no—
267C23—22no—
268C23#88611yescharacteristic
269Q8—42no—
270Q8—121no—
271Q8—121no—
272Q8—122no—
273D4—42no—
274D4—41no—
275D4—41no—
276D4—81no—
277D4—82no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4#67⋊C411yescharacteristic
288C2×C4#67⋊C411yescharacteristic
289C2×C4#88211yescharacteristic
290C2×C4C2×(C2×D3)⋊C411yescharacteristic
291C2×C4#93911yescharacteristic
292C2×C4#93911yescharacteristic
293C8—81no—
294C8—81no—
295C8—121no—
296C8—122no—
297C8—121no—
298C8—122no—
299C2×C3—41no—
300C2×C3—82no—
301C2×C3#188111yescharacteristic
302C2×C3#190811yescharacteristic
303C2×C3#191111yescharacteristic
304C22—21no—
305C22—21no—
306C22—21no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22#495811yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—42no—
317C4—241no—
318C4—241no—
319C4#520011yescharacteristic
320C4#520411yescharacteristic
321C3#937311yescharacteristic, complemented, Sylow
322C2—41no—
323C2—82no—
324C2#7007111yescharacteristic
325C2#7029311yescharacteristic
326C2#7030411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1551C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1551D3
hypocenterC3#9373
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1551

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #70815C2semidirect
5. #70429C2semidirect
6. #67757C2semidirect
7. #67708C2semidirect
321. C3#9373semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#833192
#832802
C3×#15512
#708152
#704292
#677572
#677082
#93733